Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C28 |
| direct product | C3×#34529 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C25 |
| commutator | C22 | C24×C3×C4 |
| Frattini, cosocle | C23 | C25×C3 |
| prime radical | C2×C3×H(3,𝔽4) | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
2488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2,3)(4,5,7,6)(8,9)(10,14)(11,12)(13,15)(16,17,21,22)(18,20,19,23);(9,12)(13,14)(17,22)(18,19);(4,5,7,6)(16,18,21,19)(17,23,22,20);(8,9)(10,13)(11,12)(14,15)(16,19,21,18)(17,20,22,23);(4,6,7,5)(8,10,11,15)(9,13,12,14)(16,20)(17,18)(19,22)(21,23)228 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 27517);